Optimal. Leaf size=113 \[ \frac{(d+e x)^{-2 p} (a e+c d x) \left (\frac{c d (d+e x)}{c d^2-a e^2}\right )^p \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^p \, _2F_1\left (p,p+1;p+2;-\frac{e (a e+c d x)}{c d^2-a e^2}\right )}{c d (p+1)} \]
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Rubi [A] time = 0.0940342, antiderivative size = 113, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 37, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.108, Rules used = {679, 677, 70, 69} \[ \frac{(d+e x)^{-2 p} (a e+c d x) \left (\frac{c d (d+e x)}{c d^2-a e^2}\right )^p \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^p \, _2F_1\left (p,p+1;p+2;-\frac{e (a e+c d x)}{c d^2-a e^2}\right )}{c d (p+1)} \]
Antiderivative was successfully verified.
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Rule 679
Rule 677
Rule 70
Rule 69
Rubi steps
\begin{align*} \int (d+e x)^{-2 p} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^p \, dx &=\left ((d+e x)^{-2 p} \left (1+\frac{e x}{d}\right )^{2 p}\right ) \int \left (1+\frac{e x}{d}\right )^{-2 p} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^p \, dx\\ &=\left (\left (a d e+c d^2 x\right )^{-p} (d+e x)^{-2 p} \left (1+\frac{e x}{d}\right )^p \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^p\right ) \int \left (a d e+c d^2 x\right )^p \left (1+\frac{e x}{d}\right )^{-p} \, dx\\ &=\left (\left (a d e+c d^2 x\right )^{-p} (d+e x)^{-2 p} \left (\frac{c d^2 \left (1+\frac{e x}{d}\right )}{c d^2-a e^2}\right )^p \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^p\right ) \int \left (a d e+c d^2 x\right )^p \left (\frac{c d^2}{c d^2-a e^2}+\frac{c d e x}{c d^2-a e^2}\right )^{-p} \, dx\\ &=\frac{(a e+c d x) (d+e x)^{-2 p} \left (\frac{c d (d+e x)}{c d^2-a e^2}\right )^p \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^p \, _2F_1\left (p,1+p;2+p;-\frac{e (a e+c d x)}{c d^2-a e^2}\right )}{c d (1+p)}\\ \end{align*}
Mathematica [A] time = 0.0439543, size = 97, normalized size = 0.86 \[ \frac{(d+e x)^{-2 p-1} \left (\frac{c d (d+e x)}{c d^2-a e^2}\right )^p ((d+e x) (a e+c d x))^{p+1} \, _2F_1\left (p,p+1;p+2;\frac{e (a e+c d x)}{a e^2-c d^2}\right )}{c d (p+1)} \]
Antiderivative was successfully verified.
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Maple [F] time = 1.261, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( ade+ \left ( a{e}^{2}+c{d}^{2} \right ) x+cde{x}^{2} \right ) ^{p}}{ \left ( ex+d \right ) ^{2\,p}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x\right )}^{p}}{{\left (e x + d\right )}^{2 \, p}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\left (c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x\right )}^{p}}{{\left (e x + d\right )}^{2 \, p}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x\right )}^{p}}{{\left (e x + d\right )}^{2 \, p}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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